0
$\begingroup$

If one is undefined, shouldn't the other be undefined, too? They are inverse functions. For instance, since we have that

$$\lim_{x \to \frac{\pi}{2}} \tan x = \text{undefined}$$

so too should $$\lim_{x \to \infty} \arctan x = \text{undefined}$$

since there is no value for which $\tan x$ takes infinity.

$\endgroup$
5
  • 1
    $\begingroup$ Where do you see it defined? $\endgroup$
    – badjohn
    May 19 '20 at 16:08
  • $\begingroup$ how do you mean? $\endgroup$
    – 666User666
    May 19 '20 at 16:08
  • $\begingroup$ You say "limit of $\arctan(x) $at x$=\infty$ defined" where do you see that? $\endgroup$
    – badjohn
    May 19 '20 at 16:11
  • 2
    $\begingroup$ The confusion may be due to how $\infty$ is used in limits. It might look as if we are treating $\infty$ as a number but limits involving $\infty$ have their own definitions which don't actually involve $\infty$. It is just a suggestive shorthand. $\endgroup$
    – badjohn
    May 19 '20 at 16:13
  • $\begingroup$ The points at which $\tan$ is undefined correspond to vertical asymptotes. When you flip the graph to get the inverse, these vertical asymptotes become horizontal. $\endgroup$
    – amd
    May 19 '20 at 22:26
4
$\begingroup$

The limit of tangent at $\frac\pi2$ is undefined, but the limits approaching from below and above are defined: $$\lim_{x\rightarrow\frac\pi2,\;x<\frac\pi2}\tan(x)=+\infty, \\ \lim_{x\rightarrow\frac\pi2,\;x>\frac\pi2}\tan(x)=-\infty.$$ In particular, if we restrict the function $\tan$ to the open interval $(-\frac\pi2,\frac\pi2)$, then $\tan$ has limits at both ends.

The important thing to know is that $\arctan$ is defined as the inverse of this restricted function, not the general function on $\Bbb R$ (which is not even bijective). Therefore, nothing prevents $\arctan$ from having limits at $-\infty$ and $+\infty$, and in fact $$\lim_{x\rightarrow+\infty}\arctan(x)=\frac\pi2, \\ \lim_{x\rightarrow-\infty}\arctan(x)=-\frac\pi2.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.